16.b Combinations with multiple Steps

Question 1

If there are M ways to perform task 1, and N ways to perform task 2, and these tasks are independent, how many ways are there to perform both of the tasks together?

Answer 1

M * N

Question 2

If a breakfast consists among a choice of each of 5 types of eggs, 3 types of coffees and 6 types of pastries. How many different ways could you make this breakfast?

Answer 2

These are 3 distinct groups, each choice is independent from one another.

Hence, we can prepare the breakfast in:

5 x 3 x 6 = 90 different ways

Question 3

Solve:

Answer 3

With a flip there are 2 outcomes, so…

2 x 2 x 2 x 2 x 2 = 2^5 = 32

Question 4

Solve:

Answer 4

Question 5

Set up the equation:

Answer 5

Question 6

What is a mutually exclusive event?

Answer 6

Two or more events are mutually exclusive if they cannot occur together

Question 7

If there are x ways to accomplish event A,
and y ways to accomplish event B,

And event A and B are mutually exclusive,
How would you represent the ways in which event A “OR” event B can be accomplished?

Answer 7

x + y

Question 8

What does “or” mean in a combination question

Answer 8

It very likely means that the events are mutually exclusive and must be added (rather than multiplied)

Question 9

What should you see in this question?

Answer 9

The key word “OR”
These are mutually exclusive events in this case, and thus are ADDED, NOT multiplied.

Question 10

Out of 6 different fruits at a market, we must buy at least two different types of fruit.

How would you go about solving this?

Answer 10

Consider this as 5 independent and mutually exclusive scenarios:

S1: Buy two types
S2: Buy three types
S3: Buy four types
S4: Buy five types
S5: Buy six types

Question 11

Explain the method to solving this:

Answer 11

Three different scenarios:

S1: 2i + 2t
S2: 3i + 1t
S3: 4i

find the number of ways these scenarios are possible (multiplying between), and considering they are mutually exclusive, add the number of ways the scenarios are possible.

Question 12

Express the combination formula that would arise from this problem:

(“some items must be chosen”)

Answer 12

Question 13

Express the combination formula that would arise from this problem:

(“some items must NOT be chosen”)

Answer 13

Question 14

What is the difference between:

“some items must be chosen”
AND..
“some items must NOT be chosen”

Answer 14

In questions where some items must be chosen, you subtract those items from both the group as well as the subgroup

In questions where some items must NOT be chosen, you only subtract those items from the group, but NOT from the subgroup (as you know they are not chosen anyways!)

Question 15

Solve this problem which has both items that must be and items that must NOT be chosen:

Answer 15

Question 16

What happens when events are “collectively exhaustive”?

Answer 16

Events are collectively exhaustive if, together, they represent all the potential outcomes of a situation

Question 17

In rolling a standard die, rolling an odd number and rolling an even number.. these are _____________ events, as well as ______________ events

Answer 17

In rolling a standard die, rolling an odd number and rolling an even number.. these are MUTUALLY EXCLUSIVE events, as well as COLLECTIVELY EXHAUSTIVE events

Question 18

How would you solve this?

Answer 18

By creating COLLECTIVELY EXHAUSTIVE events

Event A = 3 person committee selected consists of 3 engineers
Event B = 3 person committee selected, does NOT consist of 3 engineers

of committees NOT with 3 engineers = (total # of 3 person committees) - (# of committees with 3 engineers)

Question 19

Solve:

Answer 19

create COLLECTIVELY EXHAUSTIVE events

Question 20

Fulfill the equation:

Total number of outcomes =
(A B)

Answer 20

Total number of outcomes = (# of ways A can occur) + (# of ways B can occur)

Question 21

Set up the equation:

Answer 21

Question 22

Solve:

Answer 22

not on committee together = # of total - # of all together

Question 23

How could you solve this?

Answer 23

BUT… This is MISTAKE PRONE!

Instead, you can turn it into a mutually exhaustive event,

Question 24

Set up the equation:

Answer 24

Question 25

What should your thinking be when you see this question?

Answer 25

of clubs with at least 1 junior = # of total clubs - # of clubs with 0 juniors

Question 26

What are "dependent combinations"?

Answer 26

In which two events, A or B, are dependent if the number of ways in which event B can be selected depends on a specific way event A is selected

Question 27

What is important to see here:

Answer 27

That as you make the groups, the number of people you can choose from (n) becomes smaller!